"Three Dimensions of Space"?--D'Arcy Thompson Letter to Whitehead

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<p><strong>D’ARCY WENTWORTH THOMPSON</strong></p>


With researchers earlier this year at several universities---University of Southampton (UK), University of Waterloo (Canada), Perimeter Institute (Canada), INFN, Lecce (Italy) and the University of Salento (Italy)--- publishing findings in the journal Physical Review Letters that we live in a 2-D, i.e., a holographic universe, it was interesting to come across biologist/mathematician D’Arcy Wentworth Thompson’s 1918 letter to philosopher/mathematician Alfred North Whitehead in a recent exhibition of Thompson’s work at the University of Dundee in which Thompson addresses the issue of “Three Dimensions of Space.” Thompson and Whitehead had developed a rapport some years earlier, as students at Cambridge.

The three-page typewritten letter was loaned to the Dundee exhibit by the University of St Andrews (the two schools Thompson called home). According to University of Dundee museum curator Matthew Jarron, who organized the three-day event at Dundee and St Andrews commemorating the 100th anniversary of the publication of Thompson’s book On Growth and Form, Whitehead’s response to Thompson unfortunately has not survived.

Thompson’s letter follows.

“St Andrews.


My dear Whitehead,

I have been thinking, or dreaming, lately over a matter of which I know so little in my waking hours that I have little right to dream of it o’nights. But, like a good fellow, take me as seriously as you can, and give me the benefit of your opinion. It is nothing short of the problem of the Three Dimensions of Space.

Putting as shortly as possible what I have in mind, the points are these. And, by the way, the first difficulty seems to me, to be to decide whether there are, in reality, three dimensions of space; and the second question is, Whether there be, in reality, three dimensions or no, how did we come to think, or to find out, that it is so.

I suggest, that we are guided, even fundamentally guided, by the influence of Gravity; in other words, that in this world of ours we are always face to face with a vertical axis, and with a plane (or apparently plane) surface perpendicular to it. In other words, the right angle assumes a very special importance, and, consciously or unconsciously, we refer everything in space to trihedral coordinates.

Now suppose, on the other hand, that we were of so minute a size (or lived in a medium so dense) that gravity would have no sensible hold upon us; and suppose, owing to our minute size, that we were mainly under the influence of other, say molecular, forces. Then, to begin with, we should know nothing about a vertical, and care nothing about a right angle. And suppose, in the next place, that we lived in some sort of ‘close-packed’ or crystalline medium, say a tetrahedral one, we should never dream of three-dimensional space (unless perhaps after long mathematical investigation), but we should automatically refer everything to tetrahedral coordinates. In short, we should solemnly believe that we lived in a four-dimensional space.

So, paradox or no paradox, I seem to be driven to the conclusion that there is a quibble, or even a fallacy, underlying our definition of Space, or of Dimensions, (or perhaps both). Perhaps that ‘dimensions’


are not necessarily rectangular: or that perpendicularity, inter se, is not a fair condition to postulate of them. That the Space which actually exists is quite independent of dimensions; and that, by appropriate transformations we may ascribe to it as many as we please.

Of course, you may then say at once that even my quasi-molecular mathematician would learn, in time, that three-dimensions would suffice for his purposes, i.e. if he saw any good reason for using as few as possible. Well and good, but does that in any way prove that we have a right to say there are, in reality, three dimensions; is it anything more than a mathematical figment, an elegant formula.

It is something like Helmholtz’s theory of Colour-vision: where H. showed that we account for all our phenomena by postulating three fundamental colours, or three fundamental sets of sensory cells. Nobody doubts it; but the physiologists have been ever since wondering, and arguing whether there actually are these three: in short, whether because this is the simplest possible hypothesis, it is necessarily the true explanation.

Another point that comes into my head, with reference to the case in general, is an old and simple saying of Tait’s. He used to say that, given a symmetrical individual in symmetrical space, how on earth could you ever teach him what right and left meant. He would obviously have no right and left and space itself has, obviously, no right and left. And so, I come back to my query. Has Space really three dimensions; or is this only a convenient figment of terrestrial, and large and clumsy, mathematicians?

I put this, very much more briefly, to Larmor,- who simply swept me aside, with a story that ‘If you fix a body in space by two points, it will always come back, at length by rotation, to the same position as before’. I can’t see, for the life of me, that this answers my point. I think I have read in Clifford, of elliptical space in which it doesn’t follow in the same way. It seems to me that even if I fix it by only one point, then given perfect freedom otherwise, I may safely assert that it will still come back to the self-same position if I give it time enough.


Besides, I don’t know how to fix my body by two points in space. It reminds me of a friend of mine who wrote, and printed an article in Mind containing the remarkable phrase, ‘Let us lay down some fixed line in space, as for instance a line from London to York’. . . . . .

Kindest regards to your Wife and you. I hope to look you up some time next month.

Ever yours faithfully,